The Direct Integration Theorem: A Rigorous Framework for Consistent Discrete Solutions of the Inverse Radon Problem

The differentiable shift-variant filtered backprojection (SV-FBP) framework enables data-driven estimation of redundancy weights for cone-beam CT reconstruction under general source trajectories, removing the need for analytically derived weighting schemes. In this work, we present a systematic study of the robustness and adaptability of differentiable SV-FBP under challenging acquisition settings. We show that the framework remains stable across highly irregular and discontinuous trajectories, indicating that reconstruction performance is largely insensitive to trajectory ordering or continuity. Instead, the spatial distribution of sampling points plays a more dominant role. Under sparse-view conditions, differentiable SV-FBP achieves competitive reconstruction quality while providing an order-of-magnitude reduction in computation time compared to iterative reconstruction methods at moderate sampling densities. However, we identify a clear transition regime under severe undersampling, where the absence of iterative data consistency leads to performance degradation. Furthermore, we demonstrate that the framework remains applicable to non-planar multi-isocenter geometries, such as Lissajous-saddle trajectories, without requiring architectural modifications. These findings provide new insights into the behavior and limitations of the differentiable SV-FBP model and highlight it as a flexible and efficient solution for non-standard and robotic CBCT acquisition scenarios.

Limited-angle digital breast tomosynthesis (DBT) reconstructs a volume from a few low-dose projections over a narrow arc. At a representative nine-view, $25^{\circ}$ protocol more than 98% of image space is unmeasured, so a learned prior must supply structure in the missing wedge. Conditional diffusion priors achieve strong perceptual quality here but leave three clinical obstacles: inexact data consistency, unlocalized hallucination, and uncalibrated uncertainty. We enforce measurements exactly by replacing the per-step proximal update of a conditional diffusion sampler with exact Euclidean projection onto the data-consistent set, computed via an $m$-dimensional dual system with a one-time Gram matrix $AA^{\top}$ factorization. This projection costs 4.5 ms per step (a $248\times$ speedup) and drives the data residual to the double-precision floor ($2.4\times10^{-13}$). We prove it is the $ρ\to0$ limit of the proximal step, provide a no-harm theorem, and show that exactly consistent sample ensembles have variance supported on null($A$). Thus, the mean's entire error lies in the unmeasured subspace covered by the uncertainty map. On patient-derived breast phantoms, this improves fidelity at no depth-resolution cost. Conversely, a proximal step applied post-update degrades quality, isolating the consistency step's placement as decisive. Isotonic recalibration brings the ensemble spread to a calibrated error scale (expected calibration error $0.029\to0.008$; standardized error $4.7\to0.96$), ranking errors better than the pure prior. We also repair a 20.3% adjoint mismatch in a deployed projector via a materialized operator of record. This is the first data-consistent, uncertainty-calibrated learned reconstruction for limited-angle DBT. The solver naturally relaxes to discrepancy-ball and maximum-a-posteriori modes for noisy measurements.

Generative models, particularly Diffusion Models (DM), have shown strong potential for Computed Tomography (CT) reconstruction serving as expressive priors for solving ill-posed inverse problems. However, diffusion-based reconstruction relies on Stochastic Differential Equations (SDEs) for forward diffusion and reverse denoising, where such stochasticity can interfere with repeated data consistency corrections in CT reconstruction. Since CT reconstruction is often time-critical in clinical and interventional scenarios, improving reconstruction efficiency is essential. In contrast, Flow Matching (FM) models sampling as a deterministic Ordinary Differential Equation (ODE), yielding smooth trajectories without stochastic noise injection. This deterministic formulation is naturally compatible with repeated data consistency operations. Furthermore, we observe that FM-predicted velocity fields exhibit strong correlations across adjacent steps. Motivated by this, we propose an FM-based CT reconstruction framework (FMCT) and an efficient variant (EFMCT) that reuses previously predicted velocity fields over consecutive steps to substantially reduce the number of Neural network Function Evaluations (NFEs), thereby improving inference efficiency. We provide theoretical analysis showing that the error introduced by velocity reuse is bounded when combined with data consistency operations. Extensive experiments demonstrate that FMCT/EFMCT achieve competitive reconstruction quality while significantly improving computational efficiency compared with diffusion-based methods. The codebase is open-sourced at https://github.com/EFMCT/EFMCT.